I'll consider adial with any orientation in a place with any latitude. I'll calculate the altitude theta of the polar style on the dial, the sub-style line and its hour angle, I'll make this calculation with any of the methods known to the gnomonists.
Above this dial I may hang two threads, parallel to the dial, themselves orthogonal, whit one of the two thread directed like the sub-style line. Of course the two threads have different height.
I call substyle thread the one with this direction and hs its height above the dial. The other one is the equinoctial threads because its direction is like the equinoctial line, this thread has an he height aboce the dial.
The two threads cross themselves on a line orthogonal to the dial which touches the dial in P.

Now I may imagine to put a polar style on the dial which edge is in contact with one of the two threads. If I move the imaginary polar style, parallel to itself, keeping it in contact with the thread, its base draws a line on the dial parallel to the thread. The base line of the substyle thread is still the sub-style, the base line of the equinoctial threads is a line r orthogonal to the sub-style and crossing it in a point C which distance from P is expressed by 1).

At any istant H (gnomonic hour angle: the hour angle corresponding to the difference with the sub-style hour angle.) ao any day with declination of the Sun delta, both the polar styles will cast their shadows with the same angle H' with the sub-style direction. The shadows lengths will de different, I call bs the shadow of the polar style touching the substyle thread and be the other one.
Because the polar styles touch their thread in a casual point, moving the polar styles the edges of the two shadows draws a line parallel to the thread ant to the base line of the thread. The shadow of the substyle thread will be at a distance x from the sub-style as described by 2). In the same way the shadow of the equinoctial thread will be at a distance y from the line r as described by 3).

Now I can draw a line from C to the point where the shadows of the threads cross themselves; these line forms an angle omega with the sub-style and its tangent is the ratio between x andy as described by 4).

The ratio showed by 4) is a between two shadows length and it is the same between the length of the two polar styles, or better, it is the same between the two orthostyle: that is the height of the threads. This is an intuitive statement because the length of the shadow is directly related to the length of the style and it is demonstrable as by 5) where beta is the angle between the polar style and an hour line.

The ratio between the sine and the cosine of H' may be changed in well-known formula to calculate the angle H', the angle between the sub-style and an hour line, with its gnomonic hour angle H, as by 6).

The formula in 6) shows the direction of a generic hour line with center in C is not related to the declination of the Sun but only to the hour angle H, this means that C is the center of the bifilar sundial.

Now I can conclude the formula 6) is like the formula which describes the shadows movement on a sundial with a polar style. Exactly to get the formula in 7) I have to set the formula in 8) that is the height of the threads may be setted so that the bifilar sundial become equivalent to a polar style sundial with an altitude theta'. Setting theta' = 90° the hour lines will be 15° one from another like in an equinoctial sundial; this condition is described by 9).